Theorem bj-eldiag 37798

Description: Characterization of the elements of the diagonal of a Cartesian square. Subsumed by bj-elid6 37792. (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
bj-eldiag	⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))

Proof of Theorem bj-eldiag

Step	Hyp	Ref	Expression
1		bj-diagval2 37797	. . 3 ⊢ (𝐴 ∈ 𝑉 → (Id‘𝐴) = ( I ∩ (𝐴 × 𝐴)))
2	1	eleq2d 2849	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ 𝐵 ∈ ( I ∩ (𝐴 × 𝐴))))
3		elin 3922	. . 3 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)))
4		ancom 465	. . 3 ⊢ ((𝐵 ∈ I ∧ 𝐵 ∈ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ))
5		bj-elid4 37790	. . . 4 ⊢ (𝐵 ∈ (𝐴 × 𝐴) → (𝐵 ∈ I ↔ (1st ‘𝐵) = (2nd ‘𝐵)))
6	5	pm5.32i 584	. . 3 ⊢ ((𝐵 ∈ (𝐴 × 𝐴) ∧ 𝐵 ∈ I ) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))
7	3, 4, 6	3bitri 300	. 2 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵)))
8	2, 7	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (Id‘𝐴) ↔ (𝐵 ∈ (𝐴 × 𝐴) ∧ (1st ‘𝐵) = (2nd ‘𝐵))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∩ cin 3905   I cid 5557   × cxp 5661  ‘cfv 6538  1^st c1st 7985  2^nd c2nd 7986  Idcdiag2 37794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 7987  df-2nd 7988  df-bj-diag 37795

This theorem is referenced by: (None)